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Scaling Theorem

The scaling theorem (or similarity theorem) says that if you horizontally ``stretch'' a signal by the factor $ \alpha $ in the time domain, you ``squeeze'' and amplify its Fourier transform by the same factor in the frequency domain. This is an important general Fourier duality relationship:



Theorem: For all continuous-time functions $ x(t)$ possessing a Fourier transform,

$\displaystyle \zbox {\hbox{\sc Stretch}_\alpha(x) \;\longleftrightarrow\;\left\vert\alpha\right\vert\hbox{\sc Stretch}_{(1/\alpha)}(X)}
$

where

$\displaystyle \hbox{\sc Stretch}_{\alpha,t}(x) \isdefs x\left(\frac{t}{\alpha}\right)
$

and $ \alpha $ is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:

$\displaystyle \zbox {x\left(\frac{t}{\alpha}\right) \;\longleftrightarrow\; \left\vert\alpha\right\vert\cdot X(\alpha\omega)}$ (3.41)



Proof: See §B.4.

The scaling theorem is fundamentally restricted to the continuous-time, continuous-frequency (Fourier transform) case. The closest we come to the scaling theorem among the DTFT theorems (§2.3) is the stretch (repeat) theorem (page [*]). For this and other continuous-time Fourier theorems, see Appendix B.


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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